We study the propagator of a nonrelativistic, noninteracting particle in any nonrelativistic ``time-machine'' spacetime of the following type: an external, chronal spacetime in which two spatial regions ${\mathit{V}}_{\mathrm{\ensuremath{-}}}$ at time ${\mathit{t}}_{\mathrm{\ensuremath{-}}}$ and ${\mathit{V}}_{+}$ at time ${\mathit{t}}_{+}$ are connected by two temporal wormholes, one leading from the past side of ${\mathit{V}}_{\mathrm{\ensuremath{-}}}$ to the future side of ${\mathit{V}}_{+}$ and the other from the past side of ${\mathit{V}}_{+}$ to the future side of ${\mathit{V}}_{\mathrm{\ensuremath{-}}}$. We express the propagator explicitly in terms of those for the chronal spacetime and for the two wormholes; and from that expression we show that the propagator satisfies completeness and unitarity in the initial and final ``chronal regions'' (regions without closed timelike curves) and its propagation from the initial region to the final region is unitary. However, within the time machine it satisfies neither completeness nor unitarity. We also give an alternative proof of initial-region-to-final-region unitarity based on a conserved current and Gauss's theorem. This proof can be carried over without change to most any nonrelativistic time-machine spacetime and it is valid as long as the particle is not interacting with itself or any other quantum particle; it can, however, interact with an external field (gravitational or otherwise). This result is the nonrelativistic version of a theorem by Friedman, Papastamatiou, and Simon, which says that for a free scalar field quantum-mechanical unitarity follows from the fact that the classical evolution preserves the Klein-Gordon inner product.
We study the propagator of a nonrelativistic, noninteracting particle in any nonrelativistic time-machine'' spacetime of the following type: an external, chronal spacetime in which two spatial regions ${\mathit{V}}_{\mathrm{\ensuremath{-}}}$ at time ${\mathit{t}}_{\mathrm{\ensuremath{-}}}$ and ${\mathit{V}}_{+}$ at time ${\mathit{t}}_{+}$ are connected by two temporal wormholes, one leading from the past side of ${\mathit{V}}_{\mathrm{\ensuremath{-}}}$ to the future side of ${\mathit{V}}_{+}$ and the other from the past side of ${\mathit{V}}_{+}$ to the future side of ${\mathit{V}}_{\mathrm{\ensuremath{-}}}$. We express the propagator explicitly in terms of those for the chronal spacetime and for the two wormholes; and from that expression we show that the propagator satisfies completeness and unitarity in the initial and final chronal regions'' (regions without closed timelike curves) and its propagation from the initial region to the final region is unitary. However, within the time machine it satisfies neither completeness nor unitarity. We also give an alternative proof of initial-region-to-final-region unitarity based on a conserved current and Gauss's theorem. This proof can be carried over without change to most any nonrelativistic time-machine spacetime and it is valid as long as the particle is not interacting with itself or any other quantum particle; it can, however, interact with an external field (gravitational or otherwise). This result is the nonrelativistic version of a theorem by Friedman, Papastamatiou, and Simon, which says that for a free scalar field quantum-mechanical unitarity follows from the fact that the classical evolution preserves the Klein-Gordon inner product.
Nonrelativistic quantum theory of noninteracting particles in the spacetime containing a region with closed time-like curves (time-machine spacetime) is considered with the help of the path-integral technique. It is argued that, in certain conditions, a sort of superselection may exist for evolution of a particle in such a spacetime. All types of evolution are classified by the number n defined as the number of times the particle returns back to its past. It corresponds also to the topological class [Formula: see text] of trajectories of a particle. The evolutions corresponding to different values of n are noncoherent. The amplitudes corresponding to such evolutions may not be superposed. Instead of one evolution operator, as in the conventional (coherent) description, evolution of the particle is described by a family Un of partial evolution operators. This is done in analogy with the formalism of quantum theory of measurements, but with essential new features in the dischronal region (the region with closed time-like curves) of the time-machine spacetime. Partial evolution operators Un are equal to integrals Kn over the classes of paths [Formula: see text] if the evolution begins and ends in the chronal regions. If the evolution begins or/and ends in the dischronal region, the integral Kn over the class [Formula: see text] should be multiplied by a certain projector to give the partial evolution operator Un. Thus defined partial evolution operators possess the property of generalized unitarity ∑nUn†Un=1 and multiplicativity Um(t″, t′)Un(t′, t)=Um+n(t″, t). In the last equation however one of the numbers m or n (or both) must be equal to zero. Therefore, the part of evolution containing repeated returning backward in time cannot be factorized: all backward passages of the particle have to be considered as a single act, that cannot be presented as gradually, step by step, passing through “causal loops.” The (generalized) multiplicativity and unitarity take place for arbitrary time intervals including (i) propagating in initial and final chronal regions (containing no time-like closed curves) or from the initial chronal region to the final one, and (ii) propagating within the time machine (in the dischronal region), from the time machine to the final chronal region or from the initial chronal region to the time machine.
The Feynman propagator used in the conventional in-out formalism in quantum field theory is not a causal propagator as wave packets are propagated virtually instantaneously outside the causal region of the initial state. We formulate a causal in-out formalism in quantum field theory by making use of the Wheeler propagator, the time ordered commutator propagator, which is manifestly causal. Only free scalar field theories and their first quantization are considered. We identify the real Klein Gordon field itself as the wave function of a neutral spinless relativistic particle. Furthermore, we derive a probability density for our relativistic wave packet using the inner product between states that live on a suitably defined Hilbert space of real quantum fields. We show that the time evolution of our probability density is governed by the Wheeler propagator, such that it behaves causally too.
Spacetime must be foliable by spacelike surfaces for the quantum mechanics of matter fields to be formulated in terms of a unitarily evolving state vector defined on spacelike surfaces. When a spacetime possesses nonchronal regions which cannot be foliated by spacelike surfaces, as in the case of spacetimes with closed timelike curves, a more general formulation of quantum mechanics is required. In such generalizations the transition matrix between alternatives on two spacelike surfaces lying in regions of spacetime where foliating families can be defined may be nonunitary if a nonchronal region lies between them. This paper describes a sum-over-histories generalized quantum mechanics whose probabilities consistently obey the rules of probability theory even in the presence of such nonunitarity. The usual notion of state on a spacelike surface is lost in this generalization. Anomalies such as nonconservation of energy or "Everett phones" that are exhibited by some generalizations of quantum mechanics are not found in this one. However, the generalization is acausal in the sense that the existence of nonchronal regions of spacetime in the future can affect the probabilities of alternatives today and signaling outside the light cone is possible. The detectability of nonunitary evolution and violations of causality in measurement situations are briefly considered.
The paper deals with a recent systematic study of the propagation of relativistic quantum particles in spacetime. This study was a reaction to the overwhelming number of experiments dealing with the localization of not only massive but also of photons by detectors. The method of study is based on a configuration unitarity expansion of the vacuum-to-vacuum transition amplitudes as, massive and massless, particles propagate between emitters and detectors. Topics treated are the amplitudes of propagation from one time-space coordinate to another, limiting velocities of particles and their reconciliations with relativity, emergence of particles into cones in detection regions versus the direction of their moments, stimulated emissions by external sources in spacetime, scattering theory in quantum field theory in configuration space, and finally a spacetime for mulation of closed-time path for multi-particle states.
Quantum mechanics is traditionally formulated using the reversible coordinate time t, which enforces a globally reversible unitary group structure and time‑symmetric propagators. This conflicts with physically irreversible processes such as decoherence, measurement, entropy production, causal propagation, and curvature‑driven focusing in general relativity. We resolve this mismatch by replacing t with the monotonic temporal field T, previously developed in the context of global causal structure, temporal focusing, and worldline ordering in curved spacetime. Once T is taken as the fundamental evolution parameter, quantum dynamics cannot be reversible: the inverse operator U−1 is not physically realizable, and unitary evolution becomes a forward‑directed semigroup. We formulate **forward unitarity**, in which physical evolution is generated only by forward‑directed operators Uphys(T2,T1) with T2>T1. Pure states evolve according to a forward Schrödinger equation with an intrinsic, monotonic decoherence functional D(T), while mixed states evolve under a forward Lindblad structure. Probability is conserved, but coherent recoverability is forbidden by geometric irreversibility, causal structure, and temporal focusing. We construct a Forward‑QM Lagrangian, a forward Heisenberg picture, and a forward path integral restricted to histories with non‑decreasing T, showing how operator algebras, propagators, correlation functions, and creation/annihilation operators inherit the arrow of time. Forward‑QM provides a unified, forward‑directed formulation of quantum mechanics compatible with general relativity, causal geometry, and the irreversible structure of spacetime. It replaces reversible unitary groups with physically meaningful semigroup evolution, aligns quantum dynamics with measurement, decoherence, and entropy production, and establishes the foundation for Forward Quantum Field Theory (FQFT). The extension to FQFT—covering forward propagators, interacting fields, renormalization in the presence of monotonic decoherence, and causal compatibility in curved spacetime—will be developed in future work.
Non-Hermitian PT symmetric Klein-Gordon fields in Lorentz-violating wormholes spacetime background
We investigate a PT-symmetric Klein--Gordon (KG) oscillator in a (3+1)-dimensional Lorentz-violating (LV) traversable wormhole spacetime. The geometry is characterized by a smooth throat of radius a and a constant lapse function, A(x)=1, so that no Killing horizons occur and the two asymptotic regions at x→±∞ remain connected through the throat. The Lorentz violation modifies the radial geometry and the oscillator scale according to Ω=Ω/1−ζ. A nonminimal non-Hermitian coupling, Ft(x)=iΩx, leads to a PT-symmetric KG oscillator with a regular effective potential at the throat and a quadratic confinement at large ∣x∣. The radial equation can be mapped to the confluent Heun equation, and polynomial truncation gives a conditionally exactly solvable sector in which the oscillator frequency, throat radius, LV parameter, angular momentum, and radial quantum number satisfy algebraic constraints. For the degree-1 sector, these conditions restrict the allowed angular momenta and correlate them with the LV parameter. The resulting bound-state energies are real and discrete, with exact particle--antiparticle symmetry, E−=−E+, while the corresponding wave functions are regular and localized on the two-sided wormhole geometry. These results show how Lorentz violation and the PT-symmetric coupling modify the relativistic bound-state spectrum through the geometry of the wormhole.